Second-Order Boundary Value Problem with Integral Boundary Conditions

This paper is concerned with the existence of solutions for the second-order boundary value problem

(1.1)

where is a given function and is an integrable function.

Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [1–9] and the references therein. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example [10–14]. The goal of this paper is to give existence and uniqueness results for the problem (1.1). Our approach here is based on the Banach contraction principle and the Leray-Schauder alternative [15].

In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. Let be the space of differentiable functions whose first derivative, , is absolutely continuous.

We take to be the Banach space of all continuous functions from into with the norm

(2.1)

and we let denote the Banach space of functions that are Lebesgue integrable with norm

(2.2)

Definition 2.1.

A map is said to be -Carathéodory if

(i) is measurable for each

(ii) is continuous for almost each

(iii)for every there exists such that

(2.3)

Definition 3.1.

A function is said to be a solution of (1.1) if satisfies (1.1).

In what follows one assumes that One needs the following auxiliary result.

Lemma 3.2.

. Let . Then the function defined by

(3.1)

is the unique solution of the boundary value problem

(3.2)

where

(3.3)

Proof.

Let be a solution of the problem (3.2). Then integratingly, we obtain

(3.4)

Hence

(3.5)

(3.6)

where

(3.7)

Now, multiply (3.6) by and integrate over , to get

(3.8)

Thus,

(3.9)

Substituting in (3.6) we have

(3.10)

Therefore

(3.11)

Set Note that

(3.12)

Our first result reads

Theorem 3.3.

Assume that is an -Carathéodory function and the following hypothesis

(A1) There exists such that

(3.13)

holds. If

(3.14)

then the BVP (1.1) has a unique solution.

Proof.

Transform problem (1.1) into a fixed-point problem. Consider the operator defined by

(3.15)

We will show that is a contraction. Indeed, consider Then we have for each

(3.16)

Therefore

(3.17)

showing that, is a contraction and hence it has a unique fixed point which is a solution to (1.1). The proof is completed.

We now present an existence result for problem (1.1).

Theorem 3.4.

Suppose that hypotheses

(H1) The function is an -Carathéodory,

(H2) There exist functions and such that

(3.18)

are satisfied. Then the BVP (1.1) has at least one solution. Moreover the solution set

(3.19)

is compact.

Proof.

Transform the BVP (1.1) into a fixed-point problem. Consider the operator as defined in Theorem 3.3. We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps.

Step 1 ( is continuous).

Let be a sequence such that in Then

(3.20)

Since is -Carathéodory and then

(3.21)

Hence

(3.22)

Step 2 ( maps bounded sets into bounded sets in ).

Indeed, it is enough to show that there exists a positive constant such that for each one has .

Let . Then for each , we have

(3.23)

By (H2) we have for each

(3.24)

Then for each we have

(3.25)

Step 3 ( maps bounded set into equicontinuous sets of ).

Let , and be a bounded set of as in Step 2. Let and we have

(3.26)

As the right-hand side of the above inequality tends to zero. Then is equicontinuous. As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem we can conclude that is completely continuous.

Step 4 (A priori bounds on solutions).

Let for some . This implies by that for each we have

(3.27)

Then

(3.28)

If we have

(3.29)

Thus

(3.30)

Hence

(3.31)

Set

(3.32)

and consider the operator From the choice of , there is no such that for some As a consequence of the nonlinear alternative of Leray-Schauder type [15], we deduce that has a fixed point in which is a solution of the problem (1.1).

Now, prove that is compact. Let be a sequence in , then

(3.33)

As in Steps 3 and 4 we can easily prove that there exists such that

(3.34)

and the set is equicontinuous in hence by Arzela-Ascoli theorem we can conclude that there exists a subsequence of converging to in Using that fast that is an -Carathédory we can prove that

(3.35)

Thus is compact.

We present some examples to illustrate the applicability of our results.

Example 4.1.

Consider the following BVP

(4.1)

Set

(4.2)

We can easily show that conditions (A1), (3.14) are satisfied with

(4.3)

Hence, by Theorem 3.3, the BVP (4.1) has a unique solution on .

Example 4.2.

Consider the following BVP

(4.4)

Set

(4.5)

We can easily show that conditions (H1), (H2) are satisfied with

(4.6)

Hence, by Theorem 3.4, the BVP (4.4) has at least one solution on . Moreover, its solutions set is compact.